Problem: $ F = \left[\begin{array}{rr}1 & 3 \\ -1 & 2 \\ -2 & 1\end{array}\right]$ $ E = \left[\begin{array}{rr}-2 & -1 \\ 2 & -1\end{array}\right]$ What is $ F E$ ?
Answer: Because $ F$ has dimensions $(3\times2)$ and $ E$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ F E = \left[\begin{array}{rr}{1} & {3} \\ {-1} & {2} \\ \color{gray}{-2} & \color{gray}{1}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{-1} \\ {2} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{3}\cdot{2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{3}\cdot{2} & ? \\ {-1}\cdot{-2}+{2}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{3}\cdot{2} & {1}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{-2}+{2}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{-2}+{3}\cdot{2} & {1}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{-2}+{2}\cdot{2} & {-1}\cdot\color{#DF0030}{-1}+{2}\cdot\color{#DF0030}{-1} \\ \color{gray}{-2}\cdot{-2}+\color{gray}{1}\cdot{2} & \color{gray}{-2}\cdot\color{#DF0030}{-1}+\color{gray}{1}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}4 & -4 \\ 6 & -1 \\ 6 & 1\end{array}\right] $